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Differential dynamic microscopy (DDM) is a form of video image analysis that combines the sensitivity of scattering and the direct visualization benefits of microscopy. DDM is broadly useful in determining dynamical properties including the intermediate scattering function for many spatiotemporally correlated systems. Despite its straightforward analysis, DDM has not been fully adopted as a routine characterization tool, largely due to computational cost and lack of algorithmic robustness. We present a comprehensive statistical framework that aims at quantifying error, reducing the computational order and enhancing the robustness of DDM analysis. We quantify the error, and propagate an independent noise term to derive a closed-form expression of the expected value and variance of the observed image structure function. Significantly, we propose an unbiased estimator of the mean of the noise in the observed image structure function, which can be determined experimentally and significantly improves the accuracy of applications of DDM. Furthermore, through use of Gaussian Process Regression (GPR), we find that predictive samples of the image structure function require only around 1% of the Fourier Transforms of the observed quantities. This vastly reduces computational cost, while preserving information of the quantities of interest, such as quantiles of the image scattering function, for subsequent analysis. The approach, which we call DDM with Uncertainty Quantification (DDM-UQ), is validated using both simulations and experiments with respect to accuracy and computational efficiency, as compared with conventional DDM and multiple particle tracking. Overall, we propose that DDM-UQ lays the foundation for important new applications of DDM, as well as to high-throughput characterization.
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The problem of estimating certain distributions over ${0,1}^d$ is considered here. The distribution represents a quantum system of $d$ qubits, where there are non-trivial dependencies between the qubits. A maximum entropy approach is adopted to reconstruct the distribution from exact moments or observed empirical moments. The Robbins Monro algorithm is used to solve the intractable maximum entropy problem, by constructing an unbiased estimator of the un-normalized target with a sequential Monte Carlo sampler at each iteration. In the case of empirical moments, this coincides with a maximum likelihood estimator. A Bayesian formulation is also considered in order to quantify posterior uncertainty. Several approaches are proposed in order to tackle this challenging problem, based on recently developed methodologies. In particular, unbiased estimators of the gradient of the log posterior are constructed and used within a provably convergent Langevin-based Markov chain Monte Carlo method. The methods are illustrated on classically simulated output from quantum simulators.
We have developed a lab work module where we teach undergraduate students how to quantify the dynamics of a suspension of microscopic particles, measuring and analyzing the motion of those particles at the individual level or as a group. Differential Dynamic Microscopy (DDM) is a relatively recent technique that precisely does that and constitutes an alternative method to more classical techniques such as dynamics light scattering (DLS) or video particle tracking (VPT). DDM consists in imaging a particle dispersion with a standard light microscope and a camera. The image analysis requires the students to code and relies on digital Fourier transform to obtain the intermediate scattering function, an autocorrelation function that characterizes the dynamics of the dispersion. We first illustrate DDM on the textbook case of colloids where we measure the diffusion coefficient. Then we show that DDM is a pertinent tool to characterize biologic systems such as motile bacteria i.e.bacteria that can self propel, where we not only determine the diffusion coefficient but also the velocity and the fraction of motile bacteria. Finally, so that our paper can be used as a tutorial to the DDM technique, we have joined to this article movies of the colloidal and bacterial suspensions and the DDM algorithm in both Matlab and Python to analyze the movies.
Microscopic dynamics reveal the origin of the bulk rheological response in complex fluids. In model systems particle motion can be tracked, but for industrially relevant samples this is often impossible. Here we adapt differential dynamic microscopy (DDM) to study flowing highly-concentrated samples without particle resolution. By combining an investigation of oscillatory flow, using a novel echo-DDM analysis, and steady shear, through flow-DDM, we characterise the yielding of a silicone oil emulsion on both the microscopic and bulk level. Through measuring the rate of shear-induced droplet rearrangements and the flow velocity, the transition from a solid-like to liquid-like state is shown to occur in two steps: with droplet mobilisation marking the limit of linear visco-elasticity, followed by the development of shear localisation and macroscopic yielding. Using this suite of techniques, such insight could be developed for a wide variety of challenging complex fluids.
The quantum multiparameter estimation is very different from the classical multiparameter estimation due to Heisenbergs uncertainty principle in quantum mechanics. When the optimal measurements for different parameters are incompatible, they cannot be jointly performed. We find a correspondence relationship between the inaccuracy of a measurement for estimating the unknown parameter with the measurement error in the context of measurement uncertainty relations. Taking this correspondence relationship as a bridge, we incorporate Heisenbergs uncertainty principle into quantum multiparameter estimation by giving a tradeoff relation between the measurement inaccuracies for estimating different parameters. For pure quantum states, this tradeoff relation is tight, so it can reveal the true quantum limits on individual estimation errors in such cases. We apply our approach to derive the tradeoff between attainable errors of estimating the real and imaginary parts of a complex signal encoded in coherent states and obtain the joint measurements attaining the tradeoff relation. We also show that our approach can be readily used to derive the tradeoff between the errors of jointly estimating the phase shift and phase diffusion without explicitly parameterizing quantum measurements.
Particle size is a key variable in understanding the behaviour of the particulate products that underpin much of our modern lives. Typically obtained from suspensions at rest, measuring the particle size under flowing conditions would enable advances for in-line testing during manufacture and high-throughput testing during development. However, samples are often turbid, multiply scattering light and preventing the direct use of common sizing techniques. Differential dynamic microscopy (DDM) is a powerful technique for analysing video microscopy of such samples, measuring diffusion and hence particle size without the need to resolve individual particles while free of substantial user input. However, when applying DDM to a flowing sample, diffusive dynamics are rapidly dominated by flow effects, preventing particle sizing. Here, we develop flow-DDM, a novel analysis scheme that combines optimised imaging conditions, a drift-velocity correction and modelling of the impact of flow. Flow-DDM allows a decoupling of flow from diffusive motion that facilitates successful particle size measurements at flow speeds an order of magnitude higher than for DDM. We demonstrate the generality of the technique by applying flow-DDM to two separate microscopy methods and flow geometries.
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